From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices

Abstract

A set $P = H \cup \{w\}$ of $n+1$ points in general position in the plane is called a wheel set if all points but w are extreme. We show that for the purpose of counting crossing-free geometric graphs on such a set P, it suffices to know the frequency vector of P. While there are roughly $2^n$ distinct order types that correspond to wheel sets, the number of frequency vectors is only about $2^{n/2}$. We give simple formulas in terms of the frequency vector for the number of crossing-free spanning cycles, matchings, triangulations, and many more. Based on that, the corresponding numbers of graphs can be computed efficiently. In particular, we rediscover an already known formula for w-embracing triangles spanned by H. Also in higher dimensions, wheel sets turn out to be a suitable model to approach the problem of computing the simplicial depth of a point w in a set H, i.e., the number of w-embracing simplices. While our previous arguments in the plane do not generalize easily, we show how to use similar ideas in $\mathbb {R}^d$ for any fixed d. The result is an $O(n^{d-1})$ time algorithm for computing the simplicial depth of a point w in a set H of n points, improving on the previously best bound of $O(n^d\log n)$. Based on our result about simplicial depth, we can compute the number of facets of the convex hull of $n=d+k$ points in general position in $\mathbb {R}^d$ in time $O(n^{\max \{\omega ,k-2\}})$ where $\omega \approx 2.373$, even though the asymptotic number of facets may be as large as $n^k$.